David Cameron at prime minister's questions, where he defended the decision not to refer Jeremy Hunt to Sir Alex Allan for investigation. Photograph: PA

David Cameron has declared he has the support of the independent adviser on ministers' interests for his decision not to refer the culture secretary, Jeremy Hunt, for investigation over his handling of News Corporation's bid for the broadcaster BSkyB.

Under renewed pressure over the issue at prime minister's questions on Wednesday, Cameron produced letters between himself and the adviser, Sir Alex Allan, who his critics say should be asked to make an independent judgment over Hunt's actions and those of his special adviser Adam Smith, who resigned.

Allan's reply to the PM's letter – both sent and received on Wednesday morning – said: "I note your decision in relation to Jeremy Hunt's adherence to the ministerial code which is of course a matter for you. The fact that there is an ongoing judicial inquiry probing and taking evidence under oath means that I do not believe I could usefully add to the facts in this case, though I remain available should circumstances change or new evidence emerge."

The letters did not appease all the PM's critics, however, with opponents saying they wanted Allan not to find facts but to pass judgment on whether or not Hunt had broken the code, in particular because of the many text messages and phone calls revealed by the Leveson inquiry which show Smith kept in regular contact with News Corp over the progress of the bid. The bid for the share of BSkyB it did not already own was launched in 2010 but withdrawn late last year after the scale of the phone-hacking scandal at the News of the World, one of the company's newspapers, emerged. Smith resigned over the issue in April.

After question time, a Labour source said: "This is a David Cameron smokescreen. It's not a particularly effective one because the questions are about David Cameron's judgment and his judgment is shown to be failing."

Cameron has repeatedly defended his decision not to refer Hunt to Allan for investigation, saying it is not necessary because the Leveson inquiry is looking into the facts surrounding Hunt and Smith's dealings with News Corp, and that final judgement on whether or not to sanction a minister is made by the PM. "It's the job of the prime minister to make a judgment about the ministerial code and I have made that judgement," Cameron said again on Wednesday.

Cameron also tried to play down the decision by the Liberal Democrats, the Conservatives' partner in the coalition government, not to support them in a vote on Wednesday afternoon following a debate about Hunt's conduct which has been called by Labour.

"What we're talking about here is the relationship Conservative politicians and, frankly, Labour politicians have had over the last 20 years with News Corporation, News International and the rest of it," said Cameron. "To be fair to the Lib Dems they didn't have that relationship. Their abstention today is to make that point, and I understand that – that's politics."